Abstract

We introduce a notion of measuring scales for quantum Abelian gauge systems. At each measuring scale a finite dimensional affine space stores information about the evaluation of the curvature on a discrete family of surfaces. Affine maps from the spaces assigned to finer scales to those assigned to coarser scales play the role of coarse graining maps. This structure induces a continuum limit space which contains information regarding curvature evaluation on all piecewise linear surfaces with boundary. The evaluation of holonomies along loops is also encoded in the spaces introduced here; thus, our framework is closely related to loop quantization and it allows us to discuss effective theories in a sensible way. We develop basic elements of measuretheory on the introduced spaces which are essential for the applicability of the framework to the construction of quantum Abelian gauge theories.

Received 17 January 2011Accepted 31 March 2012Published online 07 May 2012

Acknowledgments:

J.Z. acknowledges partial support from CONACyT Grant No. 80118.

Article outline:I. INTRODUCTIONII. HOLONOMY EVALUATIONS FOR EFFECTIVE THEORIESIII. HOLONOMY EVALUATION AND CURVATURE EVALUATIONIV. COARSE GRAINING MAPS AND THE CONTINUUM LIMITV. GAUSSIAN KINEMATICAL MEASURESVI. A CUBICAL SEQUENCE OF SCALES AND CONVERGENCEVII. EVALUATION OF THE CURVATURE IN THE CONTINUUMVIII. INDEPENDENCE OF THE CHOICE OF SEQUENCE OF SCALESIX. OUTLOOK: CONSTRUCTION OF PHYSICAL THEORIES